 Regular article
 Open Access
Tracing patterns and shapes in remittance and migration networks via persistent homology
 Paul Samuel P. Ignacio^{1, 2}Email authorView ORCID ID profile and
 Isabel K. Darcy^{1}
 Received: 6 February 2018
 Accepted: 20 December 2018
 Published: 5 January 2019
Abstract
Pattern detection in network models provides insights to both global structure and local node interactions. In particular, studying patterns embedded within remittance and migration flow networks can be useful in understanding economic and sociologic trends and phenomena and their implications both in regional and global settings. We illustrate how topoalgebraic methods can be used to detect both local and global patterns that highlight simultaneous interactions among multiple nodes, giving a more holistic perspective on the network fabric and a higher order description of the overall flow structure of directed networks. Using the 2015 Asian net migration and remittance networks, we build and study the associated directed clique complexes whose topological features correspond to specific flow patterns in the networks. We generate diagrams recording the presence, persistence, and perpetuity of patterns and show how these diagrams can be used to make inferences about the characteristics of migrant movement patterns and remittance flows.
Keywords
 Migration network
 Remittance network
 Persistent homology
1 Introduction
Migration and remittances have become important facets of modern human society. The United Nations’ Department of Economics and Social Affairs (UNDESA) reports that the total number of international migrants, defined as foreignborn or foreign citizens for the purpose of estimation, has continued to grow rapidly from year 2000 to year 2015 at a rate faster than that of the world’s population. Many published reports, such as UNDESA’s [1] and World Bank’s [2], provide comprehensive statistical descriptions and forecasts on world migration and remittances. These, however, tend to focus on statistics about direct and specific regionregion interactions: origin and destination of highest flow, rate of increase or decrease of flow, etc. On the other hand, studying migration and remittances in the network setting allows us to generate a clearer picture of the complex interactions embedded in its structure. Fagiolo and Mastrorillo [3] used a complexnetwork perspective to study the binary and weighted architecture of the international migration network, exploring link and node statistics, assortativity, and clustering among others. In [4, 5], centrality in the migration network was studied in terms of weighted and unweighted vertex indegree^{1} and outdegree. Simply put, these measure the diaspora and reach of migrants coming in and going out of a particular country. In addition, community detection via link density was also explored.
Ordered list of Asian countries with abbreviation codes
Order  Country  Abbrev.  Order  Country  Abbrev. 

1  Afghanistan  AF  26  Lebanon  LB 
2  Armenia  AM  27  Malaysia  MY 
3  Azerbaijan  AZ  28  Maldives  MV 
4  Bahrain  BA  29  Mongolia  MN 
5  Bangladesh  BD  30  Myanmar  MM 
6  Bhutan  BT  31  Nepal  NP 
7  Brunei Darussalam  BN  32  Oman  OM 
8  Cambodia  KH  33  Pakistan  PK 
9  China  CN  34  Philippines  PH 
10  Hong Kong  HK  35  Qatar  QA 
11  Macau  MO  36  Republic of Korea  KR 
12  Cyprus  CY  37  Saudi Arabia  SA 
13  Dem. People’s Rep. of Korea  KP  38  Singapore  SG 
14  Georgia  GE  39  Sri Lanka  LK 
15  India  IN  40  State of Palestine  PS 
16  Indonesia  ID  41  Syria  SY 
17  Iran  IR  42  Tajikistan  TJ 
18  Iraq  IQ  43  Thailand  TH 
19  Israel  IL  44  TimorLeste  TI 
20  Japan  JP  45  Turkey  TR 
21  Jordan  JO  46  Turkmenistan  TM 
22  Kazakhstan  KZ  47  United Arab Emirates  AE 
23  Kuwait  KW  48  Uzbekistan  UZ 
24  Kyrgyzstan  KG  49  Vietnam  VN 
25  Laos  LA  50  Yemen  YE 
Our approach utilizes a relatively new method in data analysis known as persistent homology whose early roots trace back to Size Theory in the 90’s [6–9] (for connected components) and later generalized to higher dimensions by Edelsbrunner, Letscher, and Zomorodian [10]. We refer the reader to [11–13] for a detailed account of the history and development of the theory. In this note, we apply persistent homology to data viewed as a directed network, from which we build a sequence of mathematical objects each contained in the next. Distinct (nonhomologous) generators of topological features that persist significantly along the sequence of mathematical objects are then regarded as encoding the overall shape of the pointcloud. We make precise what we mean by generators in Sect. 3.2. Several works have employed this approach in studying the topology of undirected networks [14–17]. An extension of this approach to directed networks using path complexes has been explored by Chowdhury and Mémoli [18, 19], and via neighborhood complex by Horak et al. [20]. In [17], Petri et al. introduced a filtration that allowed them to compute the persistent homology of weighted undirected networks, and remarked that their methods are amenable to extension to directed networks following the directed clique construction of Palla et al. [21]. Reimann et al. [22] used the same directed clique construction to compute homology of neural networks, but they did not analyze the persistence of features. The contribution of our work is the implementation and development of Petri et al.’s idea to extend persistent homology theory to directed networks via cliques. The integration of the directional component to the theory allows us to talk about cycle types and perpetuity of topological features of directed networks. We expound on these notions in Sect. 3. We also present a modification to barcodes, the output diagram of persistent homology, by introducing a coloring scheme that distinguishes features of the same dimension.
The rest of the paper is organized as follows. Section 2 describes the data sets and the estimation measures used. In Sect. 3, we discuss the theoretical foundations, highlight the contributions of our work, and present an example using a toy network. Section 4 presents the patterns and shapes extracted from the Asian net migration and remittance networks. Finally, Sect. 5 presents a summary and concluding remarks.
2 Dataset description
According to UNDESA’s international migration report in 2015 [1], nearly half of all international migrants worldwide were born in Asia, and nearly a third live there. From year 2000 to year 2015, this region had more new international migrants than any other major area.^{2} Interestingly, Asia also ranks second on the percentage (60%) of people living in a different country but within the same region. These make Asia one of the most highly fluid regions in terms of internal migrant mobility.
We perform our analysis on the 2015 Asian net migration and remittance networks which include 50 countries and states as listed in Table 1. We use data on foreignborn population obtained from the UN Global Migration Database and on bilateral remittances from the World Bank database as reported respectively in [23] and [24].
The migration data uses estimates on foreignborn population based on censuses, registers, and models. These estimates also take into account factors such as the estimated size of the total population in the country of destination, and specific country circumstances such as sudden migrant influx or exodus due to conflict, economic booms or busts, and major changes in migration policies. For lack of data source, estimates for the Democratic People’s Republic of Korea were based on “model” countries chosen according to similarity in criterion for enumerating international migrants, proximity, and migration experience. A complete documentation of the migration data set is available in [25].
 (a)
the data on migrants in various destination countries are incomplete;
 (b)
the incomes of migrants abroad and the costs of living are both proxied by per capita incomes in terms of purchasing power parity;
 (c)
there is no way to capture remittances flowing through informal, unrecorded channels.
3 Methods
3.1 Directed clique complexes
To transform our directed network into a topological object, it is sufficient to define what the simplices are. The directed clique complex is obtained by defining simplices of dimension k to be directed \(k+1\) cliques. Doing so transfers, in a natural way, the directional information on cliques to spatial information on simplices as the latter can be visually represented by geometric objects: a 0simplex is a point, a 1simplex is a directed edge connecting two points, 2 and 3simplices are respectively filled in triangles and solid tetrahedra whose edges form directed paths from the unique source to the unique sink as in Fig. 3.
We remark here that our definition of simplices via directed cliques is motivated by the flow structure we are concerned with. A ksimplex represents a group of \(k+1\) countries where pairwise interactions exist between any two members, and the overall flow structure can be characterized by the unidirectional flow from a unique source (that sends to every country) to a unique sink (that receives from every country). Masulli and Villa [27] used the same construction to investigate how resulting topological invariants, the Euler characteristic and network degree, can be used to assess specific functional and dynamical properties of directed networks. This definition can be modified to fit an appropriate alternate model. One alternative construction, due to Grigor’yan et al. [28–30] uses paths in a directed network to define simplices. This construction allows more flexibility in the definition of simplices, which in turn gives rise to an increased complexity in simplex variety that makes the approach less intuitive. For example, the network on four vetices in Fig. 3(e) is in fact the sum of two 2paths, and hence regarded as a twodimensional simplex generator in a path complex. This however, as Chowdhury and Mémoli [19] point out, shows that path homology is able to differentiate this particular motiff from other types of cycles. Another construction due to [31] is via ordered tuple complexes where simplices are ordered tuples \((v_{0},v_{1},\ldots,v_{p})\) closed under deletion of entries, and where repetition of vertices is allowed.
3.2 Directed clique homology
In this space, any ncycle that is the boundary of a linear combination of n+1simplices is treated as zero, and hence regarded algebraically trivial. For example, the cycle \(c_{1} = [a,d] + [d,b] + [a,b]\) is trivial since it is the boundary of the 2simplex \([a,d,b]\). As a consequence, a pair of ncycles whose difference is trivial in the above sense are considered homologous. For example, the 1cycles \([a,b]+[b,c]+[c,d]+[a,d]\) in Fig. 3(f) and \([b,c]+[c,d]+[d,b]\) in Fig. 3(g) are homologous in the directed clique complex in Fig. 3(h) since their difference is precisely \(c_{1}\). Alternatively, these two 1cycles can be viewed as surrounding the same “hole” in the directed clique complex. We then say that these 1cycles generate the same class in the homology group \(H_{1}\). In a similar manner, the generators of the homology group \(H_{n}\) are the representative linear combinations of ndimensional simplices that surround distinct topological features embedded in the simplicial complex. These topological features capture complex structures and interactions in the network fabric (see Fig. 1(c)–(f)). For instance, the generators of \(H_{0}\) partition the vertices into clusters in a manner similar to single linkage clustering. We will expound on this in Sect. 3.6.
Notice that a variety of patterns may arise from a given feature. For example, a 1cycle in the directed clique complex may correspond to any of the patterns in Figs. 3(e)–3(g). For differentiation, we will refer to 1cycles whose corresponding pattern in the directed network has no source nor sink, as in Fig. 3(g), as circuits. This pattern depicts a circular flow in the network. In addition, 1cycles whose pattern has a unique source and a unique sink will be called Type 1 1cycles (see Figs. 3(e) and 3(f)), while 1cycles with multiple sources and sinks will be referred to as Type 2 1cycles.
3.3 Filtrations, persistence, and perpetuity
By examining the features of each subcomplex and keeping track of their representative generators, we can identify topological features that persist significantly across the development of the complex. Persistent cycles are regarded as significant patterns in the directed network. This is the main idea of persistent homology due to Edelsbrunner, Letscher, and Zomorodian [10] who originally applied it over arbitrary filtrations of simplicial complexes. Many adaptations and refinements have been proposed to make persistent homology computationally efficient [32, 33] and suited for a wide variety of data sets. In particular, Petri et al. [16, 17] applied persistent homology on the filtered clique complex for weighted undirected networks. As suggested in their paper, we modify their approach by adopting the directed kclique construction of Palla et al. [21].
Throughout the network development, cycles can be trivialized by becoming the boundary of higher dimensional simplices. Cycles can also become homologous to older cycles. In these cases, we say that the cycle dies. Otherwise, we say it persists. This birth and death history of homological cycles throughout the filtration can be transcribed in a barcode, a diagram consisting of bars representing the birth and death history of features of all dimensions. In a barcode, the length of the bars record the persistence of the features, and measure how long a pattern survives before it gets killed off. Figure 5(c) shows the persistence barcode of the filtered directed clique complex in Fig. 5(b): black bars represent connected components, red bars represent 1cycles, and the blue bar represents the lone 2cycle. The long black bar signifies that connectivity in the directed network is characterized by a single component. Similarly, the longest red bar detects the single persistent 1cycle (\([h,b]+[b,d]+[d,g]+[h,g]\)), and the blue bar detects the 2cycle in Fig. 4(c).
It must be noted that the intrinsic structure of the directed network and the filtration we use significantly influence how the resulting persistence barcode is read. In the usual undirected persistent homology on finite metric spaces, since the threshold filters only through distances between vertices, higher dimensional simplices are always created as the threshold scans through all distances, and holes of any dimension eventually get filled in. This means that all but one 0dimensional bar in the barcode must die. In contrast, our construction of the simplicial complex limits the simplices to maximally connected subnetworks built up from actual directed edges connecting vertices. Hence, 1dimensional simplices are introduced only when corresponding directed edges are present, and once the threshold reaches the maximum edge weight, no additional simplices can be constructed. This then means that in the barcode, the topological features of the entire directed network are encoded by the bars that never die. We will refer to these bars as perpetual. We note that perpetual bars indeed represent the Betti numbers after computing homology at the end of the filtration. As we are using a similar filtration (see Sect. 3.4) implemented by jHoles [34] to compute persistent homology for weighted undirected networks, the Betti numbers computed by jHoles and the perpetual bars in our barcode have the same interpretation.
We compute persistent directed clique homology using the standard persistence algorithm that appears in [35] adapted for directed clique complexes using code written from scratch.
3.4 Maxtomin weight filtration
We will employ this approach in computing persistent features in both net migration and remittance networks.
3.5 Variation on persistence barcodes
Since homological cycles correspond to actual subnetworks, we can encode additional information in the 1 and 2dimensional barcodes by coloring the bars according to the standard deviation of the weights in the cycles they represent. We adhere to the elder rule and use the oldest generators to represent homology classes, that is, we compute the standard deviation of all the edge weights present in the oldest linear combination of simplices that surround a topological feature. This allows differentiation between cycles of the same dimension according to the variation in the edge weights, and is most useful for distinguishing cycles that are born at a later stage in the filtration as illustrated in an example in the following section. The introduction of this coloring scheme supplements the information encoded within barcodes, and could potentially be combined with other tools, such as persistence landscapes [36, 37], homological scaffolds [38], and persistent entropy [39, 40], that enrich the information extrapolated from barcodes.
3.6 Another toy example
In this example, we will use the same techniques introduced in the beginning of this chapter. The difference from the previous example is that we are going to use the transformed values to filter the weights in the directed network so that along the filtration, edges having large weights are born first as described in Sect. 3.4.
The 2dimensional barcode detects the two spheres in Figs. 8(e) and 8(f). The first persistent bar being born before the halfway mark in the diagram shows that all edges in the 2cycle have weights bigger than half of the largest weight in the network. Although not extremely spread, the large range of values covered by its assigned color suggests that the weight distribution in the 2cycle is somewhat spread. The death of the bar signifies that this 2cycle is eventually trivialized. The perpetual black bar shows that a 2cycle with highly variable edge weights is a 2dimensional feature of the entire directed network.
4 Patterns and shapes in Asian net migration and remittance networks
Of the 61 distinct 1cycle generators throughout the Asian net migration network evolution, only 1 is a circuit (Fig. 11.55) detecting the circular flow of migrant surplus along Kazakhstan → Kyrgyztan → Tajikistan → Kazakhstan. Within this circle of migrant movement, Kazakhstan receives a net migrant flow (inflow minus outflow) of 8746 people, while Kyrgyztan and Tajikistan both lose migrants by 3256 people and 5486 people respectively. In addition, only four 1cycles of Type 1 are detected, having respective sources and sinks IN and SA (Fig. 11.32), ID and SA (Fig. 11.37), SY and JO (Fig. 11.39), and MM and BN (Fig. 11.45). The rest of the detected 1cycles are of Type 2 having developing states as sources and high income states as sinks. Moreover, while algebraically trivial circuits and Type 1 1cycles abound as meridians in many 2cycles, the grading shows that migrant flows are directed towards an external (high income) sink node. In addition, all 1cycles are born late in the filtration, and the mean standard deviation of 1cycle edge weights is very high at 81,6704 people. All these suggest a familiar scene: that migrant movement within Asia is characterized by highly unbalanced exodus of migrants from developing to high income states. That all bars in the 1dimensional barcode die at the end of the filtration reflects that there is in fact very small migration flow across states within the 1cycles other than the flow described by the 1cycle itself.
There are two perpetual 2cycles in the Asian net migration network, shown in Figs. 16(c) and 16(d). These two spheres reveal the complex flow dynamics among the countries included in the 2cycles. In particular, Fig. 16(c) shows that in this group of countries, AF and CN are sources of migrant flows while IL and MN are attractors. On the other hand, Fig. 16(d) reveals that there is movement of migrants from CN to LK via a transitory circuit that runs along BD → BT → NP → BD. As there is also one perpetual connected component, and no perpetual 1cycle, we see that the overall topology of the entire Asian migration network is characterized by two spheres (Fig. 16(c) and 16(d)) joined at a source node (CN).
There is only one 1cycle generator of Type 1 (cycle 59 in Fig. 18) having source SA and sink JO, and the rest are of Type 2 with sources mostly high income states, and sinks developing states. Similar observations as those in the migration network suggest that highly unbalanced flows of remittances from high income to developing countries abound in Asia. An interesting observation is that the second most persistent 1cycle (Fig. 19(b)) in the net remittance barcode is a copy of the most persistent 1cycle for the net migration network with all arrows reversed.
5 Summary and conclusion
In this note, we showed how persistent directed clique homology can be used to extract embedded patterns in weighted directed networks that reveal complex and simultaneous interactions among multiple vertices. Detection of these patterns affords a deeper probe into the local and global topological features of directed network models. By adopting Palla et al.’s [21] directed clique construction, we applied persistent homology to the associated directed clique complex of a weighted directed network using a filtration technique similar to the weight rank clique filtration introduced by Petri et al. [17]. The definition of directed cliques allowed us to simplify network flow structure by adopting an overall unidirectional flow in vertex communities having complete local connections.

Classifying 1cycles as circuits (no sink nor source), Type 1 (with unique sink and unique source), or Type 2 (with multiple sinks and sources). These different types of 1cycles encode different flow patterns. The definition of the directed clique also allows for the detection of the circuit with three vertices—a feature not commonly detected in other TDA methods.

Characterizing topological features of directed network by perpetuity. While persistent homology on undirected networks tracks longlived topological features along a filtration, persistent directed clique homology in addition detects topological features of the entire directed network by tracking cycle generators that never get killed off.

Distinguishing topological features of the same dimension by introducing coloring schemes for the barcodes. For the 0dimensional barcode, coloring the bars according to eventual connected component membership encodes the clustering of the vertices at the end of the filtration similar to the final clustering output of single linkage hierarchical clustering. For dimensions of at least 1, coloring the bars according to variation in the edge weights within the cycles captures information on similarity or disparity among internal flows in the cycle. It is worth noting that persistence is agnostic of such characteristic in detected cycles.
We obtained the colored barcodes after computing persistent homology from the directed clique complexes of the Asian net migration and remittance networks. For the Asian net migration network, connectivity is characterized by a single perpetual component with most linkages forming at the end of the filtration. An analysis of the sixtyone detected 1cycles shows that most are of Type 2, including the two that are most persistent. This reflects the abundance of highly unbalanced movements of migrants from groups of low income countries to high income states. Moreover, thirtyseven 2cycles are detected, all of which are spheres and two are perpetual, encoding the complex migrant flow structure among multiple countries in Asia. On the other hand, sixty 1cycles and fortytwo 2cycles are detected in the net remittance network. Most 1cycles are also of Type 2, two of which have significantly larger persistence, and all 2cycles are spheres. We generate figures for all detected 1cycles (Figs. 10–11 and 17–18) and 2cycles (Figs. 12–15) and present them in the text.
The indegree (resp. outdegree) of a vertex is the number of its incoming (outgoing) directed edges.
The regions identified by UNDESA are Africa, Asia, Europe, Latin America and the Carribean, North America, and Oceania.
A source (respectively sink) is a vertex with all incident edges pointing outward (respectively inward).
Declarations
Availability of data and materials
All data sets used in this study are available from the corresponding databases mentioned in the text. The net migration and remittance matrix is available upon request.
Funding
Paul Samuel Ignacio was supported by the University of Iowa Graduate College postcomprehensive research award for the duration of the study.
Authors’ contributions
The main idea of this paper was proposed by PSI. PSI prepared the manuscript initially, wrote all code, and prepared all figures. ID contributed ideas and to the writing of the manuscript. Both authors thoroughly read and approved the final manuscript.
Ethics approval and consent to participate
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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